Rationalizing an expression is a valuable technique when finding limits, particularly when the expression involves square roots. Here's how rationalizing helps:

How Rationalizing Helps in Finding Limits:

1. Eliminating Indeterminate Forms: In many limits involving square roots, you might encounter indeterminate forms like $\frac{0}{0}$ or $\infty - \infty$. Rationalizing helps to convert these forms into something more manageable. By multiplying by the conjugate, you often transform the indeterminate expression into a simpler form that can be evaluated directly.

2. Simplifying Complex Fractions: When the expression has both a square root term and another polynomial term, the square root might make it challenging to evaluate the limit. Multiplying by the conjugate creates a difference of squares, which eliminates the square root. This results in a polynomial or rational expression that's easier to handle.

3. Revealing Dominant Terms: At infinity, the behavior of the expression is often dominated by the highest power terms. Rationalizing helps isolate these dominant terms, making it easier to evaluate the limit by focusing on how these terms behave as the variable approaches infinity.

4. Facilitating Algebraic Manipulation: Once the square root is eliminated or simplified, other algebraic techniques, such as factoring or dividing by the highest power of $x$, become easier to apply. This simplification often reveals the limit's value directly.

Example Revisited:

In the given example:

$\lim_{x \to \infty} \left(\sqrt{x^2 + x - 1} - x\right)$

By rationalizing:

$\lim_{x \to \infty} \frac{x - 1}{\sqrt{x^2 + x - 1} + x}$

We converted an indeterminate form into a simpler expression. This allowed us to focus on the dominant behavior as $x$ approaches infinity, leading to the evaluation of the limit.

Would you like to explore another concept or need further clarification? Here are some related questions to consider:

1. What other techniques can be used to evaluate limits involving square roots besides rationalizing?
2. Can you provide an example where rationalizing doesn't work or isn't the best method?
3. How do you determine which term is dominant in a polynomial when taking limits at infinity?
4. How does rationalizing help when dealing with limits involving trigonometric functions?
5. What are the limitations of rationalizing, if any, in the context of limit problems?

Tip: When facing indeterminate forms, always consider rationalizing or other algebraic manipulations to transform the expression into a solvable form. This approach often simplifies the problem significantly.